Determination of gravity anomalies from torsion balance measurements

Transcription

1 Determnaton of gravty anomales from torson balance measurements L Völgyes G Tóth Department of Geodesy Surveyng Budapest Unversty of Technology Economcs H-5 Budapest Hungary Müegyetem rp 3 G sapó Eötvös Loránd Geophyscal Insttute of Hungary H-45 Budapest Hungary Kolumbusz u 7-3 Abstract There s a dense networ of torson balance statons n Hungary coverng an area of about m These measurements are a very useful source to study the short wavelength features of the local gravty feld especally below 30 m wavelength Our am s thus to use these exstng torson balance data n combnaton wth gravty anomales Therefore a method was developed based on ntegraton of horzontal gravty gradents over fnte elements to predct gravty anomaly dfferences at all ponts of the torson balance networ Test computatons were performed n a Hungaran area extendng over about 800 m There were 48 torson balance statons 30 ponts among them where values were nown from measurements n ths test area Keywords Gravty anomales torson balance measurements The proposed method Let s start from the fundamental equaton of physcal geodesy: T T g γ r where T s the potental dsturbance s the mean radus of the Earth (Hesanen Mortz 967 hangng of gravty anomaly g between two arbtrary ponts P s: T T r r ( g ( T T In a specal coordnate system (x ponts to North y to East z to Down the changng of gravty anomaly: T T ( g ( T T z z Let s estmate the order of magntude of term / ( T T whch s: ( γ ( T T N ( where N s the changng of geod undulaton between the two ponts If the changng of geod undulaton between two ponts s m than the value of ( s 03 mgal ( mgal 0 5 m/s Tang nto account the average dstance between the torson balance statons supposng not more than dm order of geod undulaton s changng the value of ( can be neglgble Applyng the notaton T z T / z for the partal dervatves the changng of gravty anomales between the two ponts P s: ( g ( T z ( Tz So n the case of dsplacement vector dr the elementary change of gravty anomaly wll be: d ( dr dx + dy + dz z T dx + Tdy + Tdz Integratng ths equaton between ponts P we get the changng of gravty anomaly: ( d T dx + Tdy + where Tdz ( T W T W W ; T W W are horzontal gradents of gravty measured by torson balance W s the measured vertcal gradent U U are the

2 normal value of horzontal gravty gradents U s the normal value of vertcal gradent Accordng to Torge (989: γ γ eβ U sn ϕ U 0 M U γ + + ω M N where M N s the curvature radus n the merdan n the prme vertcal γ γ ( β sn e + ϕ s the normal gravty on the ellpsod Wth the values of the Geodetc eference System 980 the followng holds at the surface of the ellpsod: U 8 sn ϕ ns U 3086 ns Let s compute the frst ntegral on the rght sde of equaton ( between the ponts P Before the ntegraton a relocaton to a new coordnate system s necessary; the connecton between the coordnate systems (xy the new one (uv can be seen on Fgure Denote the drecton between the ponts P wth u be the coordnate axs v perpendcular to u Denote the azmuth of u wth α pont the z axs to down perpendcularly to the plane of (xy (uv! Fg oordnate transformaton (xy (uv The transformaton between the two systems s: x u cosα vsnα y u snα + v cosα Usng these equatons the frst dervatves of any functon W are: + cosα + snα u u u + snα + cosα v v v From ths frst equaton f W Tz than ( T cos + T α du T dx T dy Tzu du α sn + because dx cosα du dy snα If ponts P are close to each other as requred ntegrals on the rght sde of equaton ( can be computed by the followng trapezod ntegral approxmaton formula: s ( T dx + Tdy Tzudu [( Tzu + ( T ] zu (3 h T dz [( T ( ] + T (4 where s s the horzontal dstance between ponts P h s the heght dfference between these two ponts The value of ntegral (4 depends on the vertcal gradent dsturbance T the heght dfference between the ponts If ponts are at the same heght (on a flat area n case of small vertcal gradent dsturbances the thrd ntegral n ( can be neglected (Eg the value of (4 s 05 mgal n case of h 50m [( T + ( T ]/ 50E So dscardng the effect of (4 the dfferences of gravty anomales between two ponts can be computed by the approxmate equaton: ( s { [( T + ( T ] cosα + [( T + ( T ] snα Practcal solutons } (5 If we have a large number of torson balance measurements t s possble to form an nterpolaton net (a smple example can be seen n Fgure for determnng gravty anomales at each torson balance ponts (Völgyes On the bass of Eq (5 ( g (6

3 can be wrtten between any adjacent ponts where s { ( W + ( W cosα (7 ( W + ( W + snα } Fg Interpolaton net connectng torson balance ponts For an unambguous nterpolaton t s necessary to now the real gravty anomaly at a few ponts of the networ (trangles n Fgure Let us see now how to solve nterpolaton for an arbtrary networ wth more ponts than needed for an unambguous soluton where gravty anomales are nown In ths case the g values can be determned by adjustment The queston arses what data are to be consdered as measurement results for adjustment: the real torson balance measurements W W or values from Eq (7 Snce no smple functonal relatonshp (observaton equaton wth a measurement result on one sde unnowns on the other sde of an equaton can be wrtten computaton ought to be made under condtons of adjustment of drect measurements rather than wth measured unnowns ths s however excessvely demng n terms of storage capacty Hence concernng measurements two approxmatons wll be appled: on the one h gravty anomales from measurements at the fxed ponts are left uncorrected thus they are nput to adjustment as constrants on the other h j on the left h sde of fundamental equaton (6 are consdered as fcttous measurements corrected Thereby observaton equaton (6 becomes: + v (8 permttng computaton under condtons gven by adjustng ndrect measurements between unnowns (Detreő 99 The frst approxmaton s possble snce relablty of the gravty anomales determned from measurements exceeds that of the nterpolated values consderably Valdty of the second approxmaton wll be reconsdered n connecton wth the problem of weghtng For every trangle sde of the nterpolaton net observaton equaton (8: v may be wrtten In matrx form: (9 v A x + ( m ( mn (n ( m where A s the coeffcent matrx of observaton equatons x s the vector contanng unnowns g l s the vector of constant terms m s the number of trangle sdes n the nterpolaton net n s the number of ponts The non-zero terms n an arbtrary row of matrx A are: [ ] whle vector elements of constant term l are the values Gravty anomales fxed at gven ponts modfy the structure of observaton equatons If for nstance g 0 s gven n (8 then the correspondng row of matrx A s: [ ] the changed constant term beng: j 0 that s g of coeffcents of g are mssng from vector x matrx A respectvely whle correspondng terms of constant term vector l are changed by a value g 0 Adjustment rases also the problem of weghtng Fctve measurements may only be appled however f certan condtons are met The most mportant condton s the deducblty of covarance matrx of fctve measurements from the law of error propagaton requrng however a relaton yeldng fctve measurement results n the actual case Eq (7 Among quanttes on the rght-h sde of (7 torson balance measurements W W may be consdered as wrong They are about 9 equally relable ± E ( E Eötvös Unt 0 s furthermore they may be consdered as mutually ndependent quanttes thus ther weghtng coeffcent matrx Q wll be a unt matrx Wth the nowledge of Q the weghtng coeffcent matrx Q of fctve measurements after Detreő (99 s: Q F Q l F F F

4 Q E beng a unt matrx Elements of an arbtrary row of matrx F are: n n For the followng consderatons let us produce rows f f of matrx F (referrng to sdes between ponts P P P3 respectvely: f f [ [ Usng s s 3 P P s: whle snα s s cosα snα s cosα snα3 s3 snα3 0 s3 cosα3 s3 cosα3 0 f varance of ] value referrng to sde ( sn α + cos s m α 4 s f f yeld covarance of sdes P P P3 : s s cov 4 3 ] values for ( snα snα + cosα cosα 3 3 Thus fctve measurements may be stated to be correlated the weghtng coeffcent matrx contans covarance elements at the juncton pont of the two sdes If needed the weghtng matrx may be produced by nvertng ths weghtng coeffcent matrx Practcally however two approxmatons are possble: ether fctve measurements j are consdered to be mutually ndependent so weghtng matrx s a dagonal matrx; or fctve measurements are weghted n nverted quadratc relaton to the dstance By assumng ndependent measurements the second approxmaton results also from nverson snce terms n the man dagonal of the weghtng coeffcent matrx are proportonal to the square of the sde lengths The neglecton s however justfed n addton to the smplfcaton of computaton also by the fact that contradctons are due less to measurement errors than to functonal errors of the computatonal model (Völgyes Test computatons Test computatons were performed n a Hungaran area extendng over about 800 m In the last century approxmately torson balance measurements were made manly on the flat terrtores of Hungary at present 408 torson balance measurements are avalable Locaton of these 408 torson balance observatonal ponts the ste of the test area can be seen on Fgure 3 Fg 3 Locaton of torson balance measurements beng stored n computer database the ste of the test area Fg 4 Gravty measurements (mared by dots torson balance ponts (mared by crcles on the test area The nearly flat test area can be found n the mddle of the country the heght dfference between the lowest hghest ponts s less than 0 m There were 48 torson balance statons 97 gravty measurements on ths area 30 ponts from these

5 48 torson balance statons were chosen as fxed ponts where gravty anomales g are nown from gravty measurements the unnown gravty anomales were nterpolated on the remanng 8 ponts Locaton of torson balance statons (mared by crcles the gravty measurements (mared by dots can be seen on Fgure 4 The solne map of gravty anomales g γ (γ s the normal gravty constructed from 97 g measurements can be seen on Fgure 5 Small dots ndcate the locatons of measured gravty values Measurements were made by Worden gravmeters by accuracy of ±0-30 µgal At the same tme the solne map of gravty anomales constructed from the nterpolated values from 48 torson balance measurements can be seen on Fgure 6 Small crcles ndcate the locatons of torson balance ponts the applcablty accuracy of nterpolaton we compared the gven the nterpolated gravty anomales values were determned for each torson balance ponts from gravty measurements by lnear nterpolaton on the one h gravty anomales for the same ponts from gravty gradents measured by torson balance on the other h Isolne surface maps of dfferences between the two types of values can be seen on Fgures 7 8 The dfferences are about ± mgal the maxmum dfference s 4 mgal 887EBDL KISK Fg 7 Isolne map of dfferences between the measured the nterpolated gravty anomales on the test area IZSK Fg 5 Gravty anomales from g measurements on the test area δ [mgal] Fg 8 Surface map of dfferences between the measured the nterpolated gravty anomales on the test area Fg 6 Interpolated gravty anomales from W W gradents measured by torson balance on the test area More or less a good agreement can be seen between these two solne maps In order to control Fnally the stard error characterstc to nterpolaton determned by m ± n n ( was computed (where from gravty measurements mes nt mes g s the gravty anomaly nt g s the nterpolated value from torson balance measurements n 48 s the number of torson balance statons

6 Stard error m ±8 mgal ndcates that horzontal gradents of gravty gve a possblty to determne gravty anomales from torson balance measurements by mgal accuracy on flat areas In case of a not qute flat area (le our test area accuracy of nterpolaton would probably be ncreased by tang nto consderaton the effect of vertcal gradents by ntegral (4 but unfortunately we haven t got the real vertcal gradent values of torson balance ponts on our test area yet It would be mportant to nvestgate the effect of vertcal gradent for the nterpolaton n the future Summary A method was developed based on ntegraton of horzontal gradents of gravty W W to predct gravty anomales at all ponts of the torson balance networ Test computatons were performed n a characterstc flat area n Hungary where both torson balance gravmetrc measurements are avalable omparson of the measured the nterpolated gravty anomales ndcates that horzontal gradents of gravty gve a possblty to determne gravty anomales from torson balance measurements by mgal accuracy on flat areas Accuracy of nterpolaton would probably be ncreased by tang nto consderaton the effect of vertcal gradents Acnowledgements We should than for the fundng of the above nvestgatons to the Natonal Scentfc esearch Fund (OTKA T T for the assstance provded by the Physcal Geodesy Geodynamc esearch Group of the Hungaran Academy of Scences eferences Detreő Á (99 Adjustment calculatons Tanönyvadó Budapest (n Hungaran Hesanen W Mortz H (967 Physcal Geodesy WH Freeman ompany San Francsco London Torge W (989 Gravmetry Walter de Gruyter Berln New Yor Völgyes L (993 Interpolaton of Deflecton of the Vertcal Based on Gravty Gradents Perodca Polytechnca veng Vo 37 Nr pp Völgyes L (995 Test Interpolaton of Deflecton of the Vertcal n Hungary Based on Gravty Gradents Perodca Polytechnca veng Vo 39 Nr pp Völgyes L (00 Geodetc applcatons of torson balance measurements n Hungary eports on Geodesy Warsaw Unversty of Technology Vol 57 Nr pp 03- Völgyes L Tóth Gy sapó G (004: Determnaton of gravty anomales from torson balance measurements IAG Internatonal Symposum Gravty Geod Space Mssons Porto Portugal August 30 - September Dr Lajos VÖLGYESI Department of Geodesy Surveyng Budapest Unversty of Technology Economcs H-5 Budapest Hungary Műegyetem rp 3 Web: E-mal: volgyes@ebmehu